Conservation of momentum only works because the gravitational field itself is supposed to have momentum. Similar to how momentum is conserved for an accelerating paddling boat only if you include the sea. Sea analogy is a good one I think, especially because the gravitational field interacts with itself, like water, contrary to electromagnetic fields.

Conservation laws in general are a quirky thing in GR, see: http://people.bu.edu/gorelik/ES_GG_Conservation_Laws.pdf

or http://www.preposterousuniverse.com/blog/2010/02/22/energy-i... for a shorter read

>but approximately the Earth would also accelerate faster.

Well yes, Earth has a stronger gravitational pull because of rotation. On smaller distances rotation effects become even more interesting:

https://en.wikipedia.org/wiki/Frame-dragging